Counterexamples for High-Degree Generalizations of the Schrödinger Maximal Operator
In 1980 Carleson posed a question on the minimal regularity of an initial data function in a Sobolev space that implies pointwise convergence for the solution of the linear Schrödinger equation. After progress by many authors, this was recently resolved (up to the endpoint) by Bourgain, whose counterexample construction for the Schrödinger maximal operator proved a necessary condition on the regularity, and Du and Zhang, who proved a sufficient condition. Analogues of Carleson's question remain open for many other dispersive partial differential equations. We develop a flexible new method to approach such problems and prove that for any integer, if a degree generalization of the Schrödinger maximal operator is bounded from to, then In dimensions, for every degree, this is the first result that exceeds a long-standing barrier at. Our methods are number-theoretic, and in particular apply the Weil bound, a consequence of the truth of the Riemann Hypothesis over finite fields.
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- 4902 Mathematical physics
- 0101 Pure Mathematics
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Published In
DOI
EISSN
ISSN
Publication Date
Volume
Issue
Start / End Page
Related Subject Headings
- General Mathematics
- 4902 Mathematical physics
- 0101 Pure Mathematics